Advertisement

A Study of the Efficiency of Exact Methods for Diffusion Simulation

  • Stefano Peluchetti
  • Gareth O. Roberts
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 23)

Abstract

In this paper we investigate the efficiency of some simulation schemes for the numerical solution of uni- and multi-dimensional stochastic differential equation (SDE) with particular interest in a recently developed technique for diffusion simulation [5] which avoids the need for any time-discretisation approximation (the so-called exact algorithm for diffusion simulation). The schemes considered are: the Exact Algorithm, the Euler, the Predictor-Corrector and the Ozaki-Shoji schemes. The analysis is carried out via a simulation study using some test SDEs. We also consider efficiency issues arising by the extension of EA to the multi-dimensional setting.

Keywords

Stochastic Differential Equation Acceptance Rate Exact Algorithm Discretisation Scheme Poisson Point Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Giles, Mike B.(2008) Multilevel Monte Carlo Path Simulation. In: Operations Research, vol 56, No. 3, pp 607–617.Google Scholar
  2. 2.
    Higham D. J., Mao X., Stuart A. M. (2002) Strong convergence of Euler-type methods for nonlinear stochastic differential equations. In: Journal of Numerical Analysis, vol 40, No. 3, pp 1041–1063.Google Scholar
  3. 3.
    Ait-Sahalia, Y. (2008) Closed-Form Likelihood Expansions for Multivariate Diffusions. In: Annals of Statisticss, vol 36, No. 3, pp 906–937, NBER.Google Scholar
  4. 4.
    Albanese C., Kuznetsov A. (2005) Transformations of Markov processes and classification scheme for solvable driftless diffusions. www3.imperial.ac.uk/mathfin/people/calban/papersmathfi).
  5. 5.
    Beskos A., Papaspiliopoulos O., Roberts G.O. (2006) Retrospective exact simulation of diffusion sample paths with applications. In: Bernoulli, vol 12, pp 1077–1098.Google Scholar
  6. 6.
    Beskos A., Papaspiliopoulos O., Roberts G.O. (2008) A new factorisation of diffusion measure and finite sample path construction. In: Methodology and Computing in Applied Probability, vol 10, No. 1, pp 85–104.Google Scholar
  7. 7.
    Beskos A., Roberts G.O. (2005) Exact simulation of diffusions In: Ann. Appl. Probab, vol 15, pp 2422–2444.Google Scholar
  8. 8.
    Bruno Casella (2005) Exact MC simulation for diffusion and jump-diffusion processes with financial applications. IMQ - Bocconi University.Google Scholar
  9. 9.
    Milstein G. N., Tretyakov M. V. (2004) Stochastic numerics for Mathematical Physics, Springer-Verlag New York.Google Scholar
  10. 10.
    Gilks WR (1992) Derivative-free adaptive rejection sampling for Gibbs sampling. In: Bayesian Statistics, vol 4, No. 2, pp 641–649.Google Scholar
  11. 11.
    Gilks WR, Wild P. (1992) Adaptive Rejection Sampling for Gibbs Sampling. In: Applied Statistics, vol 41, No. 2, pp 337–348, JSTOR.Google Scholar
  12. 12.
    Kloeden P.E., Platen E. (1992) Numerical Solution of Stochastic Differential Equations, Springer.Google Scholar
  13. 13.
    Maruyama G. (1955) Continuous Markov processes and stochastic equations. In: Rend. Circ. Mat. Palermo, vol 4, pp 48–90.Google Scholar
  14. 14.
    Stefano Peluchetti (2007) An analysis of the efficiency of the Exact Algorithm, IMQ - Università Commerciale Luigi Bocconi.Google Scholar
  15. 15.
    Shoji I., Ozaki T. (1998) Estimation for nonlinear stochastic differential equations by a local linearization method. In: Stochastic Analysis and Applications. vol 16, No. 4, pp 733–752, Taylor & Francis.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.HSBCLondonUK
  2. 2.Department of StatisticsUniversity of WarwickCoventryUK

Personalised recommendations